I suspect that the limit does exist as the combined power of $x$ and $y$ is higher in the numerator than in the denominator, and I have noticed a pattern where this produces a limit, but the reverse case does not.

I have tried using polar coordinates $x = r\cos{\theta}, y = r\sin{\theta}$ and simplifying to get:

I can't seem to get anywhere from here. I was trying to apply the squeezed theorem, but this expression seems like it needs to be simplified more before I can do that. Any hints on how to do that? Or am I barking up the wrong tree and need to try another approach?